reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;
reserve T, X, Y for Subset of MC-wff;
reserve p, q, r for Element of MC-wff;
reserve T, X, Y for Subset of MC-wff;
reserve p, q, r for Element of MC-wff;

theorem Th83:
  p in CnS4 (X) & p => q in CnS4 (X) implies q in CnS4 (X)
proof
  assume that
A1: p in CnS4 (X) and
A2: p => q in CnS4 (X);
  T is S4_theory & X c= T implies q in T
  proof
    assume that
A3: T is S4_theory and
A4: X c= T;
A5: p => q in T by A2,A3,A4,Def23;
    p in T by A1,A3,A4,Def23;
    hence thesis by A3,A5;
  end;
  hence thesis by Def23;
end;
